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Specifically, we show that if a tournament $G$ is \"prime\"---that is, if there is no subset $X \\subseteq V(G)$, $1 < |X| < |V(G)|$, such that for all $v \\in V(G) \\backslash X$, either $v \\rightarrow x$ for all $x \\in X$ or $x \\rightarrow v$ for all $x \\in X$---then $G$ is $U_5$-free if and only if either $G$ is a specific tournament $T_n$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1208.0398","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-02T04:37:19Z","cross_cats_sorted":[],"title_canon_sha256":"22dae2176012796696dcaafe62bd87666dea52966d339b4aa0318bf2f537bfd7","abstract_canon_sha256":"3cd51d39ff7320c6e277bfbe4c80d266244863ae054aabc20b8e864756f0b988"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:03.446134Z","signature_b64":"MP73aUGiJ8oT8zQav/a4MEkF4hVvUo7lfmTCSi//ek3x+M/DakA33P2xAIf5ZftWQRaB/x3q55yySjixIdcvDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac609d3cb7c08acaab9f2e7001484322bca9d4a0190ffb8fd606245d77081de0","last_reissued_at":"2026-05-18T02:49:03.445692Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:03.445692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structure theorem for U5-free tournaments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gaku Liu","submitted_at":"2012-08-02T04:37:19Z","abstract_excerpt":"Let $U_5$ be the tournament with vertices $v_1$, ..., $v_5$ such that $v_2 \\rightarrow v_1$, and $v_i \\rightarrow v_j$ if $j-i \\equiv 1$, $2 \\pmod{5}$ and ${i,j} \\neq {1,2}$. 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Specifically, we show that if a tournament $G$ is \"prime\"---that is, if there is no subset $X \\subseteq V(G)$, $1 < |X| < |V(G)|$, such that for all $v \\in V(G) \\backslash X$, either $v \\rightarrow x$ for all $x \\in X$ or $x \\rightarrow v$ for all $x \\in X$---then $G$ is $U_5$-free if and only if either $G$ is a specific tournament $T_n$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0398","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1208.0398","created_at":"2026-05-18T02:49:03.445759+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.0398v2","created_at":"2026-05-18T02:49:03.445759+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.0398","created_at":"2026-05-18T02:49:03.445759+00:00"},{"alias_kind":"pith_short_12","alias_value":"VRQJ2PFXYCFM","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"VRQJ2PFXYCFMVK47","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"VRQJ2PFX","created_at":"2026-05-18T12:27:25.539911+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK","json":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK.json","graph_json":"https://pith.science/api/pith-number/VRQJ2PFXYCFMVK47FZYACSCDEK/graph.json","events_json":"https://pith.science/api/pith-number/VRQJ2PFXYCFMVK47FZYACSCDEK/events.json","paper":"https://pith.science/paper/VRQJ2PFX"},"agent_actions":{"view_html":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK","download_json":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK.json","view_paper":"https://pith.science/paper/VRQJ2PFX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.0398&json=true","fetch_graph":"https://pith.science/api/pith-number/VRQJ2PFXYCFMVK47FZYACSCDEK/graph.json","fetch_events":"https://pith.science/api/pith-number/VRQJ2PFXYCFMVK47FZYACSCDEK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK/action/storage_attestation","attest_author":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK/action/author_attestation","sign_citation":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK/action/citation_signature","submit_replication":"https://pith.science/pith/VRQJ2PFXYCFMVK47FZYACSCDEK/action/replication_record"}},"created_at":"2026-05-18T02:49:03.445759+00:00","updated_at":"2026-05-18T02:49:03.445759+00:00"}